Constraint algorithm

In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are; (i) choose novel unconstrained coordinates (internal coordinates), (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods.

Constraint algorithms are often applied to molecular dynamics simulations. Although such simulations are sometimes performed using internal coordinates that automatically satisfy the bond-length, bond-angle and torsion-angle constraints, simulations may also be performed using explicit or implicit constraint forces for these three constraints. However, explicit constraint forces give rise to inefficiency; more computational power is required to get a trajectory of a given length. Therefore, internal coordinates and implicit-force constraint solvers are generally preferred.

Constraint algorithms achieve computational efficiency by neglecting the motions of intramolecular atoms. If intramolecular behavior is important, e.g. hydrogen bonding and its behaviour, constraint algorithms should not be used.

The motion of a set of N particles can be described by a set of second-order ordinary differential equations, Newton's second law, which can be written in matrix form

where M is a mass matrix and q is the vector of generalized coordinates that describe the particles' positions. For example, the vector q may be a 3N Cartesian coordinates of the particle positions r_k, where k runs from 1 to N; in the absence of constraints, M would be the 3Nx3N diagonal square matrix of the particle masses. The vector f represents the generalized forces and the scalar V(q) represents the potential energy, both of which are functions of the generalized coordinates q.

If M constraints are present, the coordinates must also satisfy M time-independent algebraic equations

where the index j runs from 1 to M. For brevity, these functions g_i are grouped into an M-dimensional vector g below. The task is to solve the combined set of differential-algebraic (DAE) equations, instead of just the ordinary differential equations (ODE) of Newton's second law.

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